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| Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.) | 
| Ref | Expression | 
|---|---|
| inimass | ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rnin 6165 | . 2 ⊢ ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) ⊆ (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶)) | |
| 2 | df-ima 5697 | . . 3 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐵) ↾ 𝐶) | |
| 3 | resindir 6013 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) | |
| 4 | 3 | rneqi 5947 | . . 3 ⊢ ran ((𝐴 ∩ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) | 
| 5 | 2, 4 | eqtri 2764 | . 2 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) | 
| 6 | df-ima 5697 | . . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶) | |
| 7 | df-ima 5697 | . . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶) | |
| 8 | 6, 7 | ineq12i 4217 | . 2 ⊢ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶)) | 
| 9 | 1, 5, 8 | 3sstr4i 4034 | 1 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∩ cin 3949 ⊆ wss 3950 ran crn 5685 ↾ cres 5686 “ cima 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 | 
| This theorem is referenced by: restutopopn 24248 | 
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